A CLASS OF METRIZABLE LOCALLY QUASI-CONVEX GROUPS

A CLASS OF METRIZABLE LOCALLY QUASI-CONVEX GROUPS WHICH ARE NOT MACKEY

arXiv:1012.5713v1 [math.GN] 28 Dec 2010

DIKRAN DIKRANJAN, ELENA MART´IN-PEINADOR, AND VAJA TARIELADZE Abstract. A topological group (G, µ) from a class G of MAP topological abelian groups will be called a Mackey group in G if it has the following property: if ν is a group topology in G such that (G, ν) ∈ G and (G, ν) has the same continuous characters, say (G, ν)∧ = (G, µ)∧ , then ν ≤ µ. If LCS is the class of Hausdorff topological abelian groups which admit a structure of a locally convex topological vector space over R, it is well-known that every metrizable (G, µ) ∈ LCS is a Mackey group in LCS. For the class LQC of locally quasi-convex Hausdorff topological abelian groups, it was proved in 1999 that every complete metrizable (G, µ) ∈ LQC is a Mackey group in LQC ([9]). The completeness cannot be dropped within the class LQC as we prove in this paper. In fact, we provide a large family of metrizable precompact (noncompact) groups which are not Mackey groups in LQC (Theorem 7.5). Those examples are constructed from groups of the form c0 (X), whose elements are the null sequences of a topological abelian group X, and whose topology is the uniform topology. We first show that for a compact metrizable group X 6= {0} the topological group c0 (X) is a non-compact complete metrizable locally quasi-convex group, which has countable topological dual iff X is connected. Then we prove that for a connected compact metrizable group X 6= {0} the group c0 (X) endowed with the product topology induced from the product X N is metrizable precompact but not a Mackey group in LQC.

1. Introduction For a locally convex space E there always exist a finest topology in the class of locally convex topologies giving rise to the same dual space E ∗ . This topology was introduced by Mackey and it was named after him. A similar setting, as we describe below, can be considered for locally quasi-convex groups, a class which properly contains that of locally convex Hausdorff spaces. However the good results known to hold for the class of locally convex spces are no longer valid in this new class. In this paper we change the point of view: starting from sufficiently large classes of abelian groups, we define the corresponding Mackey groups. As will be seen, the theory is much reacher and properties like connectedness, local compactness, etc. play an important role, which obviously was not the case in the framework of locally convex spaces. Let X, Y be groups. We denote by Hom(X, Y ) the set of all group homomorphisms from X to Y . If X, Y are topological groups, CHom(X, Y ) stands for the continuous elements of Hom(X, Y ). A set Γ ⊂ Hom(X, Y ) will be called separating, if (x1 , x2 ) ∈ X × X, x1 6= x2 =⇒ ∃γ ∈ Γ, γ(x1 ) 6= γ(x2 ) . For a topological group X, a Hausdorff group Y and a non-empty Γ ⊂ Hom(X, Y ) we denote by σ(X, Γ) the coarsest topology in X with respect to which all members of Γ are continuous. Note that σ(X, Γ) is a group topology in X; moreover, σ(X, Γ) is Hausdorff iff Γ is separating. The set S := {s ∈ C : |s| = 1} . 2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20. Key words and phrases. Mackey group, Mackey topology, precompact group, locally quasi-convex group, groups of null sequences, c0 (X), summable sequence. The authors acknowledge support of the MICINN of Spain through the grants MTM2009-14409-C02-01 for the first author and MTM2009-14409-C02-02 (subprograma MTM)for the second and third ones; V.Tarieladze was also partially supported by GNSF/ST08/3-384 and GNSF/ST09/3-120. The three authors also acknowledge to the IMI (Institute for Interdisciplinary Mathematics) for supporting several visits of Dikranjan and Tarieladze to the Complutense University of Madrid, which have allowed to carry out this joint research. 1

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DIKRAN DIKRANJAN, ELENA MART´IN-PEINADOR, AND VAJA TARIELADZE

is an abelian group with respect to multiplication of complex numbers; it is endowed with the usual topology induced from C. From now on all considered groups will be abelian. For a group G an element of Hom(G, S) is called a (multiplicative) character. Let G be a topological group. We write: G∧ := CHom(G, S) . An element of G∧ is called a continuous character. Always 1 ∈ G∧ , where 1(x) = 1 ∈ S, ∀x ∈ G. The set G∧ with respect to pointwise multiplication of characters is an abelian group with a neutral element 1. The group G∧ is called the topological dual of G. We shall not fix in advance a topology in G∧ . A topological group G is called maximally almost periodic, for short a MAP-group, if G∧ is separating. We denote by MAP also the class of all MAP-groups. For a (not necessarily discrete) topological group G the topology σ(G, G∧ ) is called the Bohr topology of G. For a group G and for a group topology τ in G we write τ + for the Bohr topology of (G, τ ). Clearly + τ ≤ τ and τ + is a Hausdorff topology iff G is MAP. Proposition 1.1. ([10]; cf. also [13, Theorem 2.3.4] and [25]) For a MAP-group (G, τ ) the following statements hold: (a) τ + is a precompact group topology. (b) (G, τ )∧ = (G, τ + )∧ . (c) (G, τ ) is precompact iff τ = τ + . Definition 1.2. Let (G, τ ) be a topological group. A group topology η in G is said to be compatible for (G, τ ) if (G, τ )∧ = (G, η)∧ . The concept of compatible group topology is due to Varopoulos [25]. This remarkable paper offers, among other results, a description of locally precompact group topologies compatible with a MAP-group (G, τ ). It is also proved there that every MAP-group (G, τ ) admits at most one locally compact group topology in G compatible with (G, τ ) [25, p. 485]. It is natural to consider the maximum (provided it exists) of all compatible topologies for (G, τ ), a topological group in a certain class of groups. The following definition is in the spirit of [4], where a categorical treatment of Mackey groups is given. Definition 1.3. Let G be a class of MAP-groups. A topological group (G, µ) will be called a Mackey group in G if (G, µ) ∈ G and if ν is a compatible group topology for (G, µ), with (G, ν) ∈ G, then ν ≤ µ. Definition 1.4. Let G be a class of MAP-groups and (G, ν) ∈ G. If there exists a group topology µ in G compatible for (G, ν) such that (G, µ) ∈ G and (G, µ) is a Mackey group in G, then µ is called the G-Mackey topology in G associated with ν. In this paper we will not discuss the problem of existence of G-Mackey topologies (cf. [12]). Let LPC be the class of Hausdorff locally precompact topological abelian groups. A precompact (G, τ ) ∈ LPC may not be a Mackey group in LPC (for instance, if (G, τ ) is an infinite discrete group, then (G, τ + ) ∈ LPC is not a Mackey group in LPC. Clearly, (G, τ ) ∈ LPC, τ is compatible for (G, τ + ), but τ is strictly finer than τ + ). However, the metrizability changes the picture as the following statement shows. Theorem 1.5. (cf. [25, Corollary 2 (p.484)]) Every metrizable (G, µ) ∈ LPC is a Mackey group in LPC. Next we consider another class of groups in which the metrizable groups are again Mackey. We say that a topological group G admits a structure of a (locally convex) topological vector space over R if there exists a map R × G → G making G a (locally convex) topological vector space over R. It is known that whenever a Hausdorff topological abelian group admits a topological vector space structure over R, it must be unique.


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Let LCS be the class of Hausdorff topological abelian groups which admit a structure of locally convex topological vector space over R. It is an important consequence of Hahn-Banach theorem that LCS ⊂ MAP. The next theorem is proved in Section 2. Theorem 1.6. Every metrizable (G, µ) ∈ LCS is a Mackey group in LCS. Remark 1.7. Let MAPVS be the class of MAP-groups which admit a structure of topological vector space over R. It is known that LCS ⊂ MAPVS and that this inclusion is strict. An analogue of Theorem 1.6 fails for MAPVS: A metrizable (G, µ) ∈ MAPVS is a Mackey group in MAPVS iff (G, µ) is locally compact (cf. [9, Proposition 2.1]). A natural class of groups containing LPC ∪ LCS is provided by LQC, the class of locally quasi-convex Hausdorff groups (see Definition 2.1). In fact, it is known that (1.1)

LPC ∪ LCS ⊂ LQC ⊂ MAP

where the inclusions are strict (cf. [2, 3, 9]). It was proved in [9] that every Cech-complete (G, µ) ∈ LQC is a Mackey group in LQC. In particular, every locally compact (G, µ) ∈ LQC is a Mackey group in LQC. We prefer to isolate here another particular case more relevant for the specific purposes of this paper: Theorem 1.8. ([9]) Every complete metrizable (G, µ) ∈ LQC is a Mackey group in LQC. In view of Theorems 1.5,1.6, and 1.8, the following question arises: Question 1.9. Is every metrizable (G, µ) ∈ LQC a Mackey group in LQC ? We provide a negative answer to this question, in fact we have: Theorem 1.10. Let (G, τ ) ∈LQC be a non-precompact group with countable dual (G, τ )∧ . Then (G, τ + ) is a metrizable precompact group which is not a Mackey group in LQC. In order to use Theorem 1.10 and produce a large scale of counter-example to Question 1.9, we need a general construction of non-precompact locally quasi-convex groups with countable dual (to which the theorem can be applied). This goal is achieved my means of a class of groups which roughly speaking are ’groups of sequences’. Let us denote by c0 (X) the subgroup of X N of all null sequences of X. The following holds: Theorem. Let X be an infinite compact metrizable abelian group, u0 the uniform topology induced from X N on c0 (X). Then G := (c0 (X), u0 ) is a nonprecompact locally quasi-convex Polish group. Further, the following assertions are equivalent: (i) X is connected. (ii) G∧ = (X ∧ )(N) . (iii) Card G∧ = ℵ0 . (iv) Card G∧ < c. To the proof of this theorem (together with other more precise results) are dedicated Sections 3–6. The key point of the proof is the introduction of a class B of groups X (see Definition 6.1), such that c0 (X)∧ = (X ∧ )(N) for every precompact group X ∈ B (Theorem 6.3). It turns out, that this new class B can be described in some cases through well-known properties. Namely, in Corollary 6.13 (resp., Theorem 6.15) we show that in the class of all locally compact (resp., metrizable) groups, X ∈ B iff X is connected (resp., locally generated in the sense of Enflo [15], see Definition 6.10). This is the backbone of the proof of the theorem. In the last section we offer some open questions and conjectures. Notation and terminology. Let S denote the circle group and S+ = {s ∈ S : Re(s) ≥ 0} . c will stand for the cardinality of continuum. We denote by e the neutral element of a group. We also use the symbols 0 and 1 instead of e if the group is known to be additive or multiplicative respectively. For a subset A of a group X denote by hAi the subgroup of X generated by A. Let X be a set. As usual, X N will denote the set of all sequences x = (xn )n∈N of elements of X and (pn )n∈N the sequence of projections X N → X.

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For a group X, X (N) will be the subgroup of X N consisting of all sequences eventually equal to e. For n ∈ N define an injective homomorphism νn : X → X (N) ,

by νn (x) = (e, . . . , e, x, e, . . . ),

where x ∈ X is placed in position n. If X is a topological group, let c0 (X) := {(xn )n∈N ∈ X N : lim xn = e }. n

Clearly c0 (X) is a subgroup of X containing X ; moreover, c0 (X) = X (N) iff X has only trivial convergent sequences. For a topological group X, N (X) is the set of all neighborhoods of e ∈ X. Clearly, S+ ∈ N (S). We ∧ write Xco for the group X ∧ endowed with the compact-open topology. For a subset V of X let: N

(N)

V ⊲ := {ξ ∈ X ∧ : ξ(V ) ⊂ S+ } . If X, Y are topological abelian groups and ϕ ∈ CHom(X, Y ), the mapping ϕ∧ : Y ∧ → X ∧ , defined by ϕ∧ (η) = η ◦ ϕ for η ∈ Y ∧ , is a group homomorphism called the dual homomorphism. The von Neumann’s kernel of a topological abelian group X is defined by \ n(X) = {ker ξ : ξ ∈ X ∧ }. Clearly, n(X) is a subgroup of X, and X is MAP iff n(X) = {0}. If n(X) = X (i.e., X ∧ = {1}), X is called minimally almost periodic. 2. Locally quasi convex groups Let us recall the definition of a locally quasi convex group. Definition 2.1. [27] A subset A of a topological group G is called quasi-convex if for every x ∈ G\A there exists χ ∈ G∧ such that χ(A) ⊂ S+ , but χ(x) 6∈ S+ . A topological group G is called locally quasi-convex if N (G) admits a basis consisting of quasi-convex subsets of G. Similar concepts were defined later in [23], where the terms polar set and locally polar group are used instead of ’quasi-convex set’ and ’locally quasi-convex group’. The author might not have been aware of [27]. The locally precompact groups are a prominent class of locally quasi-convex groups. The following statement characterizes the groups X ∈ LPC with countable duals. Proposition 2.2. For an infinite locally precompact Hausdorff topological abelian group X TFAE: (i) X is precompact metrizable. (ii) X ∧ is countable. Proof. (i) =⇒ (ii). This follows from [18, (24.14)]. (ii) =⇒ (i). Let Y be the completion of X. It is known that the groups X ∧ and Y ∧ are algebraically isomorphic, hence, Y ∧ is countable. On the other hand, Y is a locally compact Hausdorff topological ∧ abelian group therefore Yco is LCA. Since a second category countable Hausdorff topological group is ∧ ∧ ∧ discrete, we have that Yco is a discrete countable group. Hence (Yco )co is a compact metrizable group. ∧ ∧ By Pontryagin’s theorem, Y and (Yco )co are topologically isomorphic. Thus Y is compact metrizable and its topological subgroup X is precompact metrizable. Remark 2.3. It follows from Proposition 2.2 that if X is a compact non-metrizable group, then Card(X∧ ) > ℵ0 . We shall see below that implication (ii) =⇒ (i) of Proposition 2.2 may fail if X is a locally quasi-convex Hausdorff group. Proposition 2.4. Let X be a precompact Hausdorff topological group and V ∈ N (X). Then V ⊲ is a finite subset of X ∧ .


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∧ Proof. It is known that V ⊲ is a compact subset of Xco . ∧ Suppose first that X is compact. Then Xco is discrete and in this case V ⊲ is finite. If X is not compact, then it can be viewed as a dense subgroup of a compact Hausdorff topological group K. Let U denote the closure of V in K; clearly U ∈ N (K). Since S+ is closed in S, the density of V in U implies that V ⊲ = {ξ|X : ξ ∈ U ⊲ }.

Now, U ⊲ finite, implies that V ⊲ is finite as well.

Let us conclude this section with the proofs of Theorems 1.6 and 1.10. Proof of Theorem 1.6. Let (G, τ ) ∈ LCS. Then (G, τ )′ will stand for the set of τ -continuous linear forms from G to R. For every l ∈ (G, τ )′ the mapping ρl : G → S defined by ρl (x) = exp{2πil(x)} for all x ∈ G is a continuous character. It is easy to see that the mapping ρ : (G, τ )′ → (G, τ )∧ , given by l 7→ ρl is an injective group homomorphism. It is surjective as well [18, (23.32)]. Therefore, we have: (G, τ )∧ = {ρl : l ∈ (G, τ )′ } .

(2.1)

Take a metrizable (G, µ) ∈ LCS and let (G, τ ) ∈ LCS be such that (G, τ )∧ = (G, µ)∧ . Then from (2.1) ′ ′ we get (G, τ ) = (G, µ) . Since µ is a metrizable locally convex vector topology in G, the last equality according to [22, IV.3.4] implies that τ ≤ µ and we are done. 2 Proof of Theorem 1.10. Since (G, τ )∧ is countable, τ + is metrizable. The topology τ + is precompact and compatible for (G, τ ) by Proposition 1.1. The group (G, τ + ) ∈ LQC because precompact groups are locally quasi-convex. Since (G, τ ) is not precompact we have that τ + < τ being τ + 6= τ again by Proposition 1.1. Hence (G, τ + ) is a metrizable precompact group which is not a Mackey group in LQC. 2 3. Groups of sequences N

3.1. The uniform topology in X . In what follows X will be a fixed Hausdorff topological abelian group. We denote by pX the product topology in X N and by bX the box topology in X N . It is easily verified that the collection {V N : V ∈ N (X)} is a basis at e for a group topology in X N which we denote by uX and call the uniform topology. In all three cases we shall omit the subscript X when no confusion is possible. The topology u in X N is nothing else but the topology of uniform convergence on N when the elements of X N are viewed as functions from N to X and X is considered as a uniform space with respect to its left (=right) uniformity. Since it plays an important role in the sequel, we give in the next proposition an account of its main properties. We write: p0 := p|c0 (X) , b0 := b|c0 (X) and u0 := u|c0 (X) . Proposition 3.2. Let (X, +) be a Hausdorff topological abelian group. (a) The uniform topology u is a Hausdorff group topology in X N with p ≤ u ≤ b. Moreover, (a1 ) p|X (N) = u|X (N) ⇐⇒ X = {0}. (a2 ) u|X (N) = b|X (N) =⇒ X is a P-group =⇒ u = b; in particular, if X is metrizable and u|X (N) = b|X (N) , then X is discrete. (b) The passage from X to (X N , u) preserves (sequential) completeness, metrizability, MAP and local quasi-convexity. (c) If X 6= {0} , then: (c1 ) (X N , u) is not separable. (c2 ) (X (N) , u|X (N) ) is not precompact and hence, (c0 (X), u0 ) and (X N , u) are neither precompact.


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Proof. (a) The first assertion has a straightforward proof. (a1 ) Suppose that X 6= {0}. Take x ∈ X \ {0}. Then νk (x) ∈ X (N) , k = 1, 2, . . . and the sequence (νk (x))k∈N tends to 0 in p. Since X is Hausdorff, there is a V ∈ N (X) such that x 6∈ V . Then νk (x) 6∈ V N , k = 1, 2, . . . . Hence, the sequence (νk (x))k∈N does not tend to 0 in u. Q (a2 ) Suppose that u|X (N) ≥ b|X (N) . Take arbitrarily Un ∈ N (X), n = 1, 2, . . . . Then ( n∈N Un ) ∩ X (N) is a neighborhood of zero in b|X (N) . As u|X (N) ≥ b|X (N) , there is a V ∈ N (X) such that Y V N ∩ X (N) ⊂ ( Un ) ∩ X (N) . n∈N

Q . So, V ⊆ Un , n = From νk (V ) ⊆ V N ∩X (N)T, k = 1, 2, . . . , we get: νk (V ) ⊆ ( n∈N Un )∩X (N) , k = 1, 2, . . .T 1, 2, . . . , therefore V ⊂ n∈N Un . Thus, for each sequence Un ∈ N (X), n = 1, 2, . . . , n∈N Un ∈ N (X). Consequently, X is a P-group. The implication ’X is a P-group =⇒ u = b’ is easy to verify. The last assertion in (a2 ) follows from the well-known fact that a metrizable P-space is discrete. (b) We omit the standard proofs of the first two cases. Assume that X is MAP and let x = (xn )n∈N ∈ X N \ {0}. Take n ∈ N such that pn (x) 6= 0. Since X is MAP, there is ξ ∈ X ∧ such that ξ(pn (x)) 6= 1. Clearly pn is u-continuous, therefore ϕ := ξ ◦pn ∈ (X N , u)∧ and ϕ(x) 6= 1. Hence, (X N , u) is MAP. In order to prove that (X N , u) is locally quasi-convex T provided that X has the same property, just observe that for any quasi-convex V ∈ N (X), V N = n∈N p−1 n (V ) , and quasi-convexity is preserved under inverse images by continuous homomorphisms and under arbitrary intersections. (c1 ) Let us fix x ∈ X \ {0} and U ∈ N (X) such that U ∩ {−x, x} = ∅. Write C = {0, x}N . Then (3.1)

card(C) = c ,

y1 ∈ C, y2 ∈ C, y1 6= y2 =⇒ y1 − y2 6∈ U N .

Take a symmetric V ∈ N (X) such that V + V ⊂ U . Let D be a dense subset of (X N , u). Then for every y ∈ C we can find a dy ∈ D such that dy ∈ V + y. From (3.1) we get: y1 ∈ C, y2 ∈ C, y1 6= y2 =⇒ dy1 6= dy2 . Thus, for any dense subset D, card(D) ≥ c, and therefore (X N , u) is nonseparable. (c2 ) Fix x ∈ X, x 6= 0 and a symmetric V ∈ N (X) such that x 6∈ V . Then (3.2)

m, n ∈ N, m 6= n =⇒ νm (x) − νn (x) 6∈ V N .

Now (3.2) together with νm (x)−νn (x) ∈ X (N) , ∀m, n ∈ N, yield that (X (N) , u|X (N) ) is not precompact. Remark 3.3. If X is a compact metrizable group and ρ is an invariant metric for X, then the equality d∞ (x, y) = sup ρ(xn , yn ),

x, y ∈ X N

n∈N

defines an invariant metric for (X N , u). In particular, the topology of (SN , u) can be induced by the following metric: d∞ (x, y) = sup |xn − yn |, x, y ∈ SN . (†) n∈N

According to [11, Example 4.2] the metric group (SN , d∞ ) has the following remarkable property: it is not precompact, but every uniformly continuous real-valued function defined on it, is bounded. 3.4. The group of null sequences c0 (X). In this section we study the group of all null sequences of a topological abelian group X as a subgroup of (X N , u). Lemma 3.5. Let X be a topological group. Then: (a) c0 (X) is closed in (X N , u). Pn (b) If x = (xn )n∈N ∈ c0 (X), then the sequence ( k=1 νk (xk ))n∈N converges to x in the topology u; in particular, X (N) is a u-dense subset of c0 (X).


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Proof. In order to prove (a), pick x = (xn )n∈N X N \c0 (X). There exists then V ∈ N (X) and a subsequence xnk ∈ / V , k ∈ N. Take a symmetric V1 ∈ N (X) such that V1 +V1 ⊂ V . Now we have: (x+(V1 )N )∩c0 (X) = ∅. In fact, for any z ∈ (V1 )N , xnk + znP ∈ / V1 for otherwise xnk ∈ V1 + V1 ⊂ V . Therefore x + z ∈ / c0 (X). k n In order to prove (b), let yn = k=1 νk (xk ) for n ∈ N. Fix U ∈ N (X) and pick a symmetric V ∈ N (X) with V ⊂ U . Since x = (xn )n∈N ∈ c0 (X), for some k0 ∈ N we have xk ∈ V, ∀k ≥ k0 , i.e., yk − x ∈ V N ⊂ U N . Hence yk ∈ x + U N ∀k ≥ k0 and therefore yn converges to x. This proves also the last assertion of (b). Thus, the situation described in the previous Lemma is: X (N)

u−densely

⊂

c0 (X)

u−closed

⊂

XN

Since the groups of the form (c0 (X), u0 ) are the main object of our future considerations, we summarize now those properties inherited from the corresponding ones in (X N , u), or lifted from properties of X (N) . Proposition 3.6. Let X be a Hausdorff topological abelian group. (a) (c0 (X), u0 ) is a Hausdorff topological group having as a basis at zero the collection {V N ∩ c0 (X) : V ∈ N (X)}. (b) p0 ≤ u0 ≤ b0 . Moreover, p0 = u0 ⇐⇒ X = {0}; if X is metrizable and u0 = b0 , then X is discrete. (c) The passage from X to (c0 (X), u0 ) preserves (sequential) completeness, metrizability, separability, MAP, local quasi-convexity, non-discreteness, and connectedness. Proof. (c) Assume X is separable. The density of X (N) in c0 (X) yields that c0 (X) is also separable. If (c0 (X), u0 ) is discrete, for some V ∈ N (X) we have that V N ∩ c0 (X) = {0}. Thus V = {0} and X is discrete. The rest of (c) (except connectedness), as well as (a,b) follows from Proposition 3.2. Assume now that X is connected. Consequently, the product spaces X n are also connected, for all n ∈ N. Let Gn := {x ∈ c0 (X) : xk = 0, k = n + 1, n + 2, . . . } S Then (Gn , u|Gn ) is topologically isomorphic to (X n , p|X n ), therefore connected. Since X (N) = n∈N Gn T and n∈N Gn 6= ∅, we obtain that that (X (N) , u|X (N) ) and its closure (c0 (X), u0 ) are connected.

Remark 3.7. The metric group (c0 (S), d∞ ) was introduced by Rolewicz in [21], where he proves that it is a monothetic group. As he underlines, a monothetic and completely metrizable group need not be compact or discrete, a fact which breaks the dichotomy existing in the class of LCA-groups: namely, a monothetic LCA-group must be either compact or discrete ([26, Lemme 26.2 (p. 96)]; see also [1, Remark 5], where a construction of a different example of a complete metrizable monothetic non-locally compact group is indicated). A proof of the fact that (c0 (S), u0 ) is monothetic is contained in [13, pp. 20–21] (cf. also [16], where it is shown further that (c0 (S), u0 ) is Pontryagin reflexive).

Remark 3.8. Let X be the group R with the usual topology. (1) By Proposition 3.2 (RN , u) is a complete metrizable topological abelian group. The group (RN , u) is not connected; the connected component of the null element coincides with l∞ and the topology u|l∞ is the usual Banach-space topology of l∞ . It follows that although RN is a vector space over R, the topological group (RN , u) is not a topological vector space over R. (2) By Proposition 3.6 (c), (c0 (R), u0 ) is a complete separable metrizable connected topological abelian group. Note that c0 (R) is a vector space over R and (c0 (R), u0 ) is a topological vector space over R. The topology u0 is the usual Banach-space topology of c0 . (3) It is easy to see that Z(N) is a closed subgroup of (c0 (R), u0 ) and the quotient group (c0 (R), u0 )/Z(N) is topologically isomorphic with (c0 (S), u0 ).


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For an additive topological abelian group X we introduce the following three subgroups of X N included between X (N) and c0 (X): ! n X N cs(X) = {x = (xn )n∈N ∈ X : xk is a Cauchy sequence in X}, k=1

ss(X) = {x = (xn )n∈N ∈ X N :

n X

k=1

and

xk

n∈N

!

is a convergent sequence in X }. n∈N

l(X) = {x = (xn )n∈N ∈ X N : (xσ(n) )n∈N ∈ ss(X) for every bijection σ : N → N }. The same notation will be used if X is a multiplicative topological abelian group: in fact, these three groups are defined similarly. Clearly, X (N) ⊂ l(X) ⊂ ss(X) ⊂ cs(X) ⊂ c0 (X). It is easy to observe that for a Hausdorff topological abelian group X the equality ss(X) = cs(X) holds iff X is sequentially complete. The notation cs(X) and ss(X) are not standard, while l(X) can be justified as follows: usually l stands for the set of all real absolutely summable sequences and by Riemann-Dirichlet theorem, l(R) = {x = (xn )n∈N ∈ RN : (|xn |)n∈N ∈ ss(R)} = l .

(3.3)

Observe that we also have the following analogue of (3.3) for S (cf. [8, Ch. VIII.2, Theorem 1 (p.116)]): l(S) = {x = (xn )n∈N ∈ SN : (|1 − xn |)n∈N ∈ ss(R)} .

(3.4)

It is well known that l(R) 6= ss(R) 6= c0 (R). Let us consider now the situation in the general case. First of all we note that if X has only trivial convergent sequences, then X (N) = l(X) = ss(X) = cs(X) = c0 (X) . However, for a group X the equality cs(X) = c0 (X) need not imply the equality X (N) = c0 (X) as the following proposition shows. Proposition 3.9. Let X be a topological abelian group. (a) If N (X) admits a base consisting of subgroups of X, then cs(X) = c0 (X). (b) If X is sequentially complete and N (X) admits a base consisting of subgroups of X, then l(X) = ss(X) = c0 (X). (c) If X is totally disconnected and locally compact, then l(X) = ss(X) = c0 (X). Proof. (a) is easy to verify and (b) follows from (a). Finally, (c) follows from (b), since our hypothesis implies that N (X) admits a basis consisting of subgroups of X ([18, Theorem II.7.7 (p. 62)]). We shall see (Remark 6.17) that if a non-trivial group X is either connected and metrizable or connected and locally compact, then ss(X) 6= c0 (X). It is not clear whether for a complete metrizable abelian group X the equality l(X) = ss(X) implies the equality ss(X) = c0 (X). 4. The β-dual of a group of sequences For topological abelian groups X, Y and a non-empty A ⊂ X N we write: Aβ (Y ) = {h = (ξn )n∈N ∈ (CHom(X, Y ))N : (ξn (xn ))n∈N ∈ ss(Y )

∀x = (xn )n∈N ∈ A}.

β

The notation A (Y ) is taken from the theory of sequence spaces, where it is used for X = Y = R (see, e.g.,[5]). Instead of Aβ (S) we will use the shorter notation Aβ . In other words, Aβ = {h = (ξn )n∈N ∈ (X ∧ )N : (ξn (xn ))n∈N ∈ ss(S) ∀x = (xn )n∈N ∈ A}. Clearly, Aβ is a subgroup of (X ∧ )N containing (X ∧ )(N) . If A is a subgroup, we will call Aβ the β-dual of A.


9

Let A ⊂ X N be a subgroup and τ a group topology in A. The main motivation to introduce the β-dual comes from the fact that, under appropriate hypotheses, (A, τ )∧ is canonically isomorphic with Aβ . Now we describe several cases when this occurs, being A := c0 (X) the most important of them (See next section). Lemma 4.1. Let X be a topological abelian group. Then β β (4.1) X (N) = (X ∧ )N and X N = (X ∧ )(N) .

Proof. The first equality in (4.1) is trivial. To prove the second one it is sufficient to show that X N (X ∧ )(N) . Take β (4.2) h = (ξn )n∈N ∈ X N

β

⊂

and let us see that the assumption

h = (ξn )n∈N 6∈ (X ∧ )(N)

(4.3)

leads to a contradiction with (4.2). From (4.3) we get the existence of a strictly increasing sequence (kn )n∈N of natural numbers such that ξkn (X) 6= {1}, n = 1, 2, . . .

(4.4)

Since S+ contains no nontrivial subgroup of S, from (4.4) we get the existence of a sequence (xkn )n∈N of elements of X such that ξkn (xkn ) 6∈ S+ , n = 1, 2, . . .

(4.5)

Put xj = 0Qfor all j ∈ N \ {k1 , k2 , . . . }. For the sequence x = (xn )n∈N ∈ X N obtained in this way, the n sequence ( k=1 ξk (xk ))n∈N converges in S by virtue of (4.2). This implies that limn ξn (xn ) = 1. Hence, limn ξkn (xkn ) = 1. However, as S+ ∈ N (S), the equality limn ξkn (xkn ) = 1 contradicts (4.5). In the following assertion an additive abelian group A ⊂ X N is treated as a ZN -module. Lemma 4.2. Let X be a topological abelian group, A ⊂ X N a non-empty subset having the property: {0, 1}NA ⊂ A ,

(4.6)

(ξn )n∈N ∈ Aβ and (kn )n∈N a strictly increasing sequence of natural numbers. Then (ξkn )n∈N ∈ Aβ . Proof. Take an arbitrary x = (xn )n∈N ∈ A and define y = (yn )n∈N ∈ X N as follows: ykn = xn , n = 1, 2, . . . and yj = 0, ∀j ∈ N \ {k1 , k2 , . . . }. Then y = (yn )n∈N ∈ A by (4.6). Since y = (yn )n∈N ∈ A and (ξn )n∈N ∈ Aβ , the sequence ! n Y ξk (yk ) k=1

n∈N

converges in S. Observe that (4.7)

kn Y

ξj (yj ) =

n Y

ξkj (xj ), n = 1, 2, . . .

j=1

j=1

From (4.7) we conclude that the sequence  

n Y

j=1



ξkj (xj )

n∈N

converges in S too. Since this is true for an arbitrary x = (xn )n∈N ∈ A, we get that (ξkn )n∈N ∈ Aβ .

10


For a topological abelian group X, a subset A ⊂ X N and for a fixed h = (ξn )n∈N ∈ Aβ we define a mapping χh : A → S by the equality: n ∞ Y Y χh (x) = ξn (xn ) := lim ξk (xk ), x = (xn )n∈N ∈ A . n

n=1

k=1

It is easy to observe that • if h = (ξn )n∈N ∈ (X ∧ )(N) , then χh is defined on the whole X N . • If A is a subgroup of X N , then χh ∈ Hom (A, S) ,

∀h ∈ Aβ .

Notation 4.3. For a subgroup A ⊂ X N , the letter χ will denote in the sequel the mapping χ : Aβ → Hom (A, S) defined by the equality: χ(h) = χh ∀h = (ξn )n∈N ∈ Aβ . Of course, the mapping χ depends on A, but, to simplify notation, we shall not indicate this dependence. Lemma 4.4. Let X be a topological abelian group and A ⊂ X N a subgroup such that X (N) ⊂ A. Then, the mapping χ : Aβ → Hom (A, S) is an injective group homomorphism. Proof. It is easy to see that χ is a group homomorphism. Let h = (ξn )n∈N ∈ ker(χ). Then χh (x) = 1,

∀x = (xn )n∈N ∈ A.

(N)

Fix n ∈ N and x ∈ X. As νn (x) ∈ X ⊂ A, we get: ξn (x) = χh (νn (x)) = 1. Since x ∈ X is arbitrary, ξn must be the null character. Therefore, h = 1 := (1, 1, . . . ) and ker(χ) = {1}. Hence, χ is injective. The following example illustrate the usefulness of the non-topological Lemma 4.1. Example 4.5. It is a well-known fact, that if X is a topological abelian group, then ∧ (4.8) χh ∈ X N , p ∀h = (ξn )n∈N ∈ (X ∧ )(N) = (X N )β ∧ and the mapping χ : (X ∧ )(N) → X N , p is a group isomorphism. Indeed, (4.8) is easy to verify. The injectivity of χ derives from Lemma 4.4. It remains to show that χ is surjective too. Write G = X N , p and fix an arbitrary κ ∈ G∧ . We need to find h = (ξn )n∈N ∈ (X ∧ )(N) such that κ = χh . For every n ∈ N the homomorphism νn : X → G is continuous. Hence ξn := κ ◦ νn ∈ X ∧ . Let us see that h := (ξn )n∈N meets the requirements. Pn Fix x = (xn )n∈N ∈ X N . Evidently, the sequence ( k=1 νk (xk ))n∈N converges in G to x = (xn )n∈N . Hence, n n X Y (4.9) κ(x) = lim κ( νk (xk )) = lim ξk (xk ) . n

k=1

n

k=1

β Since x = (xn )n∈N ∈ X is arbitrary, (4.9) implies that h = (ξn )n∈N ∈ X N and κ = χh . By Lemma β 4.1, X N = (X ∧ )(N) . Consequently we have found h = (ξn )n∈N ∈ (X ∧ )(N) such that κ = χh , and the surjectivity of χ is thus proved. N

Remark 4.6. Let X be a topological abelian group. ∧ (a) Example 4.5 asserts only that the group X N , p can be algebraically identified with the group (X ∧ )(N) by means of the group isomorphism χ. In fact more is known: χ is also a homeomorphism ∧ between X N , p co and (X ∧ )(N) , ˆb , where ˆb stands for the topology induced from the box ∧ N product (Xco ) , b in (X ∧ )(N) . (b) An application of (a) for X = Z gives that S(N) , b|S(N) is a complete non-metrizable group. In particular, we get that S(N) is a closed subgroup of SN , b . ∧ ∧ N ) ,p . (c) It is known also that X (N) , b|X (N) co is topologically isomorphic with (Xco


11

∧ (d) An application of (c) for X = S gives that the group S(N) , b|S(N) has cardinality c. It follows that b|S(N) is not a compatible topology for S(N) , p|S(N) (cf. Proposition 5.3). 5. The topological dual of (c0 (X), u0 ). Coincidence with the β-dual In this section we will prove that for a complete metrizable gruoup X, the dual of the topological group (c0 (X), u0 ) algebraically coincides with the β-dual. We start calculating the β-dual for the particular group c0 (S), which has interest in itself: in fact, from it we derive the first example of a metrizable locally quasi-convex group which is not LQC-Mackey (See Proposition 5.3). The following statement is a bit more delicate than Lemma 4.1. Proposition 5.1. For c0 (S) we have: c0 (S)β = (S∧ )(N) .

(5.1)

Proof. For a fixed m ∈ Z let ϕm : S → S be the mapping t 7→ tm . It is known that S∧ = {ϕm : m ∈ Z}. So, fix a sequence (mn )n∈N ∈ ZN such that (ϕmn )n∈N ∈ c0 (S)β and let us see that in fact (mn )n∈N ∈ Z(N) . Suppose that (mn )n∈N 6∈ Z(N) . Then for some strictly increasing sequence (kn )n∈N of natural numbers we shall have: mkn 6= 0, n = 1, 2, . . . As (ϕmn )n∈N ∈ c0 (S)β , by Lemma 4.2 we have: (ϕmkn )n∈N ∈ c0 (S)β .

(5.2)

Let x1 = x2 = 1; then for a natural number j > 2 find the unique natural number n with 2n < j ≤ 2n+1 and write 1 xj = exp 2πi . mkj 2n+1 Clearly, x = (xj )j∈N ∈ c0 (S) and (5.3)

n+1 2Y

j=2n +1

It follows from (5.3) that



ϕmkj (xj ) = exp 2πi  

n Y

j=1

n+1 2X

k=2n +1



1  = −1, n = 1, 2, . . . 2n+1



ϕmkj (xj )

n∈N

is not a Cauchy sequence in S, hence it is not convergent in S in contradiction with (5.2).

The next proposition plays a pivotal role in the computation of the dual of (c0 (X), u0 ). Proposition 5.2. Let X be a topological abelian group and G := (c0 (X), u0 ). The following assertions hold: (a) χh ∈ G∧ for every h = (ξn )n∈N ∈ (X ∧ )(N) ⊂ c0 (X)β and the mapping χ : (X ∧ )(N) → G∧ is an injective group homomorphism. (b) We have: G∧ ⊂ {χh : h = (ξn )n∈N ∈ c0 (X)β } . (c) If X is complete metrizable (or, more generally, (c0 (X), u0 ) is a Baire space), then we have: G∧ = {χh : h = (ξn )n∈N ∈ c0 (X)β } and the mapping χ : c0 (X)β → G∧ is a group isomorphism.

12


Proof. (a) As u0 ≥ p0 we have χh ∈ G∧ , ∀h = (ξn )n∈N ∈ (X ∧ )(N) . The rest follows from Lemma 4.4. (b) Fix κ ∈ G∧ . We need to find h = (ξn )n∈N ∈ c0 (X)β such that κ = χh . For every n ∈ N the homomorphism νn : X → G is continuous, so ξn := κ ◦ νn ∈ X ∧ . Let us see that h := (ξn )n∈N meets the requirements. Pn Fix x = (xn )n∈N ∈ c0 (X). By Lemma 3.5 the sequence ( k=1 νk (xk ))n∈N converges in G to x = (xn )n∈N . Hence, n n X Y (5.4) κ(x) = lim κ( νk (xk )) = lim ξk (xk ) . n

k=1

n

k=1

Since x = (xn )n∈N ∈ c0 (X) is arbitrary, (5.4) implies that h = (ξn )n∈N ∈ c0 (X)β and κ = χh . (c) Taking into account (b), we only need to see that G∧ ⊃ {χh : h = (ξn )n∈N ∈ c0 (X)β } .

So, fix h = (ξn )n∈N ∈ c0 (X)β . As we have already noted, χh : c0 (X) → S is a group homomorphism. For n ∈ N, set hn = (ξ1 , . . . , ξn , 1, 1, . . . ). Then hn ∈ (X ∧ )(N) . Hence, χhn ∈ G∧ , n = 1, 2, . . .

(5.5) Clearly, (5.6)

lim χhn (x) = χh (x), n

∀x = (xn )n∈N ∈ c0 (X) .

Since X is complete metrizable, by Proposition 3.6(c), the group G = (c0 (X), u0 ) is complete metrizable too. In particular, G = (c0 (X), u0 ) is a Baire space. This and relations (5.5) and (5.6), according to Osgood’s theorem [19, Theorem 9.5 (pp. 86–87)] imply that the function χh has a u0 -continuity point x = (xn )n∈N ∈ c0 (X). Since χh is a group homomorphism, we get that χh is u0 -continuous. Therefore, χh ∈ G∧ . Now we are ready to give the first example of a precompact metrizable group which is not a Mackey group in LQC: Proposition 5.3. For X = S the following assertions hold: (a) (cf. [20, Lemma]) (c0 (S), p0 )∧ = (c0 (S), u0 )∧ . In particular, the set (c0 (S), u0 )∧ is countable. (b) u0 is a compatible locally quasi-convex Polish group topology for (c0 (S), p0 ). (c) (c0 (S), p0 ) is a precompact metrizable group which is not a Mackey group in LQC. Further, it is connected and monothetic. Proof. (a) Since p0 ≤ u, we have: (c0 (S), p0 )∧ ⊂ (c0 (S), u0 )∧ . To prove the converse inclusion, fix an ∧ β arbitrary κ ∈ (c0 (S), u0 ) . By Proposition 5.2(b) there exists h = (ξn )n∈N ∈ (c0 (S)) such that κ = χh . By Proposition 5.1 c0 (S)β = (S∧ )(N) . Therefore we have: κ = χh , where h = (ξn )n∈N ∈ (S∧ )(N) . Consequently, κ ∈ (c0 (S), p0 )∧ and the first part of (a) is proved. The second part of (a) follows from the first one because (c0 (S), p0 )∧ is algebraically isomorphic to (SN , p)∧ . (b) u0 is a locally quasi-convex Polish group topology by Proposition 3.6. By (a) it is compatible for (c0 (S), p0 ). (c) Observe that (c0 (S), p0 ) is a topological subgroup of the compact metrizable group (SN , p). Therefore it is metrizable and precompact. By (b), u0 is a locally quasi-convex group topology compatible for (c0 (S), p0 ) and strictly finer than p0 (by Proposition 3.6).This proves that (c0 (S), p0 ) is not a Mackey group in LQC. The last two assertions follow respectively from Proposition 3.6 (c) and from Remark 3.7. Remark 5.4. It follows from Proposition 3.6 and Proposition 5.3(a), that (c0 (S), u0 ) is a non-precompact ∧ locally quasi-convex group with countable dual (c0 (S), u0 ) ; consequently by Theorem 1.10 (c0 (S), (u0 )+ ) is a metrizable precompact group which is not a Mackey group in LQC. Observe, however that (again by Proposition 5.3(a)) we have: (u0 )+ = p0 and we get a second proof of Proposition 5.3(c). We shall see below that the group S in Proposition 5.3 can be replaced by an arbitrary non-trivial compact connected metrizable group (see Theorem 7.5). However the proof of this fact will require a subtle preparation, to which the rest of the paper is devoted.


13

6. The class B In this section we introduce a large class of compact metrizable groups X that can be used as input in Proposition 5.3. Definition 6.1. For a topological abelian group X, let Γabs (X) := {ξ ∈ X ∧ : (ξ(xn ))n∈N ∈ ss(S) ∀x = (xn )n∈N ∈ c0 (X)} . Clearly, Γabs (X) is a subgroup of X ∧ . Denote by B the class of groups X such that Γabs (X) = {1}. Remark 6.2. Let X be a topological abelian group. (a) The notation Γabs (X) is justified by the following facts easy to prove: Γabs (X) = {ξ ∈ X ∧ : (ξ(xn ))n∈N ∈ l(S),

∀x = (xn )n∈N ∈ c0 (X)} .

Taking into account the equality (3.4), for any character ξ ∈ X ∧ , we have: ξ ∈ Γabs (X) iff

∞ X

|1 − ξ(xk )| < ∞, ∀x = (xn )n∈N ∈ c0 (X).

k=1

(b) Γabs (X) can be described also by means of the diagonal homomorphism ∆ : X ∧ −→ (X ∧ )N . Clearly, if ξ ∈ X ∧ , then ξ ∈ Γabs (X) iff (ξ, ξ, . . . , ξ, . . . ) ∈ c0 (X)β . Therefore: Γabs (X) = ∆−1 (c0 (X)β ). Whenever c0 (X)β = (X ∧ )(N) , Γabs (X) = {1}, and X ∈ B. In particular, by Proposition 5.1, S ∈ B. (c) We shall show below (Theorem 6.3), that if X ∈ B is a precompact group , then (c0 (X), u0 )∧ can be canonically identified with (X ∧ )(N) . Thus, summarizing: c0 (X)β = (X ∧ )(N) =⇒ X ∈ B,

whilst

X ∈ B & X is precompact =⇒ c0 (X)∧ = (X ∧ )(N) .

Our ultimate aim to have the equality c0 (X)∧ = (X ∧ )(N) has motivated the introduction of the class B. In what follows assume that c0 (X) is endowed with the topology u0 . The dual c0 (X)∧ is a subgroup of the group Hom(c0 (X), S), and the density of X (N) in c0 (X) allows the algebraic identification of c0 (X)∧ with a subgroup of (X ∧ )N contained in c0 (X)β (see also the assignment κ 7→ (ξn ) from the proof of item (b) of Proposition 5.2). In the sequel we denote by j : G∧ → (X ∧ )N this assignment κ 7→ (ξn ), which actually is the inverse of χ defined in the previous section. Then j(G∧ ) ⊆ c0 (X)β . Theorem 6.3. If X ∈ B is precompact, and G := (c0 (X), u0 ), then j(G∧ ) = (X ∧ )(N) . Hence the mapping χ : (X ∧ )(N) → G∧ is a group isomorphism. Proof. By Proposition 5.2(a), we only need to see that χ is surjective. To this end fix h = (ξn )n∈N ∈ j(G∧ ) ⊆ c0 (X)β ⊆ (X ∧ )N . We have to see that, in fact, h ∈ (X ∧ )(N) . By using the u0 continuity of χh we can find some V ∈ N (X) such that χh V N ∩ c0 (X) ⊂ S+ . This implies: (6.1)

ξn ∈ V ⊲ ,

∀n ∈ N .

By Proposition 2.4 the precompactness of X implies that the set V ⊲ is finite. Thus, by (6.1), the set Wh := {ξ ∈ V ⊲ : ∃n ∈ N, ξ = ξn } is finite. Fix ξ ∈ Wh and write Nξ := {n ∈ N : ξ = ξn }. Clearly, Nξ 6= ∅, ∀ξ ∈ Wh . We need to prove the following

14


Claim. If the set Nξ is infinite for some ξ ∈ Wh , then ξ = 1. Proof of the Claim. Let Nξ is infinite for ξ ∈ Wh . Write Nξ as a strictly increasing sequence: Nξ = {k1 , k2 , . . . }. From (ξn )n∈N ∈ c0 (X)β by Lemma 4.2 we conclude that (ξkn )n∈N ∈ c0 (X)β . Hence, (ξ, ξ, . . . , ξ, . . . ) ∈ c0 (X)β , and so, ξ ∈ Γabs (X). As X ∈ B, we obtain that ξ = 1. This proves the claim. As the set Wh is finite, we deduce from the claim that for some n0 ∈ N we have: ξn = 1, ∀n ≥ n0 . Consequently, h = (ξn )n∈N ∈ (X ∧ )(N) and the surjectivity of χ is proved. Corollary 6.4. If X ∈ B is a non-trivial precompact group and G := (c0 (X), u0 ), then Card G∧ = Card X ∧ . Proof. According to Theorem 6.3, the mapping χ : (X ∧ )(N) → G∧ is a group isomorphism. Hence, Card G∧ = Card (X ∧ )(N) . Clearly, for any precompact nontrivial group X ∈ B, X ∧ is infinite. Otherwise X would also be finite and c0 (X) would coincide with X (N) Then Γabs (X) 6= 1, which contradicts the fact that X ∈ B. Thus, Card G∧ = Card (X ∧ )(N) = Card X ∧ . 6.5. Properties of the class B. It is clear that B contains all minimally almost periodic groups. We will consider soon other interesting examples. Proposition 6.6. Let X be a topological abelian group. (a) If cs(X) = c0 (X), then Γabs (X) = X ∧ . (b) If cs(X) = c0 (X) and X ∧ 6= {1}, then X 6∈ B. (c) If X 6= {0} is locally compact and totally disconnected, then X 6∈ B. Proof. (a) is easy to verify and (b) follows from (a). Finally, (c) follows from (b) and Proposition 3.9 (b) because X ∧ 6= {1}. The class B is stable through continuous homomorphisms (and in particular, through quotients), as proved in the next lemma.. Lemma 6.7. Let X, Y be topological abelian groups and ϕ ∈ CHom(X, Y ). We have: (a) ϕ∧ (Γabs (Y )) ⊂ Γabs (X). (b) If X ∈ B and ϕ(X) is dense in Y , then Y ∈ B. (c) If n(X) is the von-Neumann’s kernel of X, then X ∈ B iff X/n(X) ∈ B. Proof. (a) is easy to verify. (b) By the assumption Γabs (X) = {1}, and from (a) we get that ϕ∧ (Γabs (Y )) = {1}. Now it remains to note that ϕ∧ is injective by the density of ϕ(X) in Y . (c) The implication X ∈ B =⇒ X/n(X) ∈ B follows from (b). The implication X/n(X) ∈ B =⇒ X ∈ B follows from the fact that the canonical homomorphism ϕ : X → X/n(X) induces an isomorphism ϕ∧ : (X/n(X))∧ → X ∧ . Now we prove that the class B is stable also under arbitrary direct products. Proposition 6.8. Q Let I be a non-empty index set, (Xi )i∈I a family of topological groups. Then the cartesian product i∈I Xi belongs to B iff Xi ∈ B for every i ∈ I. Q Proof. Assume that Xi ∈ B for every i ∈ I. Fix ϕQ∈ ( i∈I Xi )∧ . It is known (see, e.g., [3] or [13, Exercise 2.10.4(g,h)]) that there is a family (ξi )i∈I ∈ i∈I Xi∧ such that Card{i ∈ I : ξi 6= 1} < ∞ and Y Y ϕ(x) = ξi (xi ), ∀x = (xi )i∈I ∈ Xi . i∈I

i∈I

Q Suppose now that ϕ ∈ Γabs ( i∈I Xi ). It is easy to see that ξi ∈ Γabs (Xi ), ∀i ∈ QI. From the assumption Γabs (Xi ) = {1}, ∀i ∈ I, we get that ξi = 1, ∀i ∈ I. Hence, ϕ = 1 and so, Γabs ( i∈I Xi ) = {1}. The converse follows from Lemma 6.7, (b).


15

6.9. A description of the class B. Here we offer a complete description of the metrizable groups in B (see Theorem 6.12 and Corollary 6.13). The following notion due to Enflo [15] is a cornerstone in the sequel. Definition 6.10. [15, p. 236] A topological group X is called locally generated if hV i = X

∀V ∈ N (X) .

It is easy to observe that a topological group X is locally generated iff X=

∞ [

(V + · · ·k summands + V ),

∀V ∈ N (X) .

k=1

Some easy properties and known facts of the locally generated groups are collected in the next Remark. Remark 6.11. (a) All connected topological groups are locally generated ([18, Theorem II.7.4 ]). On the other hand, every locally generated locally compact group is connected [18, Corollary II.7.9 ]. (b) Obviously, a group X is locally generated iff X has no proper open subgroups. Consequently, if H is a dense subgroup of a topological group G, then H is locally generated iff G is locally generated. (c) From (a) and (b) one can deduce that a locally precompact group is locally generated iff its completion is connected (i.e., the locally generated locally precompact groups are precisely the dense subgroups of the connected locally compact groups). (d) The additive group of rational numbers Q with the usual topology is a metrizable locally generated group which is totally disconnected. A complete metrizable locally generated topological abelian group also can be totally disconnected [15, Example 2.2.1]. Theorem 6.12. Let X be a topological abelian group. (a) If X ∈ B, then X is locally generated. (b) If X is locally generated and metrizable, then X ∈ B. Proof. (a) Take a symmetric open V ∈ N (X) and let H := hV i. Then H is open in X and so, the quotient group X/H is discrete. Now from Lemma 6.7 (b) we get that the discrete group X/H ∈ B. This implies that X/H is a singleton. Consequently, H = X and so, X is locally generated. (b) Take a character ξ ∈ X ∧ \ {1}. In order to prove that ξ ∈ / Γabs (X) we must find a sequence (xn )n∈N ∈ c0 (X) such that ! n Y ξ(xk ) k=1

n∈N

is not convergent in S. As ξ ∈ X ∧ \ {1}, there is x ∈ X such that ξ(x) 6= 1. Let {V1 , V2 , ...} be a basis for N (X) such that Vn ⊃ Vn+1 , n = 1, 2, . . . . Since X is locally generated, X=

∞ [

(Vn + · · ·k summands + Vn ),

∀n ∈ N .

k=1

Thus for a given n ∈ N we can find kn ∈ N such that x ∈ Vn + · · ·kn summands + Vn . Therefore, we can also find a finite sequence xn,1 , . . . , xn,kn such that xn,i ∈ Vn , i = 1, . . . , kn and x=

kn X i=1

xn,i .


16

Let m0 = 0, mn :=

n X

ki ,

n = 1, 2, . . .

i=1

Define now a sequence (xj )j∈N as follows: find for j ∈ N the unique n ∈ N with mn−1 < j ≤ mn and put xj := xn,j−mn−1 . Clearly, x1 = x1,1 , . . . , xm1 = x1,m1 , xm1 +1 = x2,1 , . . . , xm2 = x2,k2 , xm2 +1 = x3,1 , . . . , xm3 = x3,k3 , . . . and xj ∈ Vn , j = mn−1 + 1, . . . , mn ; n = 1, 2, 3, . . . The last relation, since mn → ∞ and (Vn )n∈N is a decreasing basis for N (X), implies that the sequence (xj )j∈N converges to zero in X. Now, ! kn kn mn X Y Y xn,i = ξ(x) 6= 1, n = 1, 2, 3, . . . ξ(xn,i ) = ξ ξ(xj ) = j=mn−1 +1

Consequently,

Q

n ξ(x ) j j=1

i=1

n∈N

i=1

is not a Cauchy sequence in S and hence it is not convergent.

Since connected groups are locally generated, we obtain: Corollary 6.13. A metrizable abelian group X ∈ B iff X is locally generated. In particular, B contains all connected metrizable groups. Remark 6.14. Since B contains all minimally almost periodic groups (see Lemma 6.7 (c)), from Theorem 6.12(a) we conclude that a minimally almost periodic group is necessarily locally generated. In [13, p. 21] this observation is used for producing a Hausdorff group topology τ in Z(N) such that (Z(N) , τ ) is minimally almost periodic. The following theorem implies, in particular, that the metrizability assumption can be removed from Theorem 6.12 (b) in the locally compact case (however this cannot be done in general, see Remark 6.16)). Theorem 6.15. For a locally compact abelian group X TFAE: (i) X ∈ B. (ii) X is locally generated. (iii) X is connected. Proof. (i) =⇒ (ii). By Theorem 6.12 (a), (i) implies that X is locally generated. (ii) =⇒ (iii). Follows from [18, Corollary (7.9)]. (iii) =⇒ (i) Take ξ ∈ Γabs (X) and let us verify that ξ = 1. Consider the set [ ϕ(R) . A= ϕ∈CHom(R,X)

Let us see first that (6.2)

ξ|A = 1 .

Fix ϕ ∈ CHom(R, X) and set H = ϕ(R). Clearly ξ|H ∈ Γabs (H). By Theorem 6.12 (b), R ∈ B. By Lemma 6.7(b), H = ϕ(R) ∈ B too. Therefore, ξ|H ∈ Γabs (H) = {1} and ξ|H = 1. Consequently (6.2) is proved. Clearly (6.2) implies: (6.3)

ξ|hAi = 1 .

Now, according to [18, Theorem 25.20 (p.410)] the connectedness of X implies that hAi is a dense subgroup of X. From this and (6.3) we obtain that ξ = 1.


17

Remark 6.16. Local compactness is essential for the implication (iii) =⇒ (i) of Theorem 6.15. Indeed, that implication may fail in general even for a pseudocompact group X. In fact, according to [17, Corollary 2.10] (see also [24, Remark 3.4]) there exists an infinite connected pseudocompact abelian group X which contains no non-trivial convergent sequence. For such a group we have that c0 (X) = X (N) ; hence Γabs (X) = X ∧ and so, X 6∈ B. Consistent examples of connected countably compact groups without infinite compact subsets can be found in [14, Corollary 2.21] (note that in both cases the groups are sequentially complete). Remark 6.17. From the proof of Theorem 6.12(b) and implication (ii) =⇒ (i) of Theorem 6.15 it follows that if a locally generated Hausdorff topological abelian group X is either complete metrizable or locally compact, then c0 (X) 6= ss(X). According to [7, Ch. III. 5, Exercise 6 (b)(p. 315)] if X 6= {0} is an arbitrary locally generated complete Hausdorff topological abelian group, then l(X) 6= ss(X) and hence, c0 (X) 6= ss(X) as well. Remark 6.16 implies that similar conclusion may fail for a connected (hence locally generated) sequentially complete Hausdorf pseudocompact abelian group. Another class of non-metrizable groups in B can be obtained from Proposition 6.8. 7. Applications of the class B 7.1. Groups with countable dual. Proposition 7.2. Let X 6= {0} be a compact abelian group and G := (c0 (X), u0 ). We have: (a) If X is connected, then Card G∧ = Card X ∧ . (b) If X is connected and metrizable, then Card G∧ = ℵ0 . (c) If X a metrizable and disconnected, then Card G∧ = c. Proof. (a) follows from Corollary 6.4 via implication (iii) =⇒ (i) of Theorem 6.15. (b) follows from Theorem 6.12 (b), and the equality CardX∧ = ℵ0 ( Proposition 2.2). (c) It is easy to verify that (7.1)

ξ ∈ Γabs (X) =⇒ {1, ξ}N ⊂ c0 (X)β .

Since X is compact metrizable, by Proposition 5.2 (c) we get: (7.2)

ξ ∈ Γabs (X) , h ∈ {1, ξ}N =⇒ χh ∈ G∧ .

Now by Theorem 6.15, X 6∈ B and consequently there exists ξ ∈ Γabs (X), with ξ 6= 1. Then Card {1, ξ}N = c. Therefore, {χh : h ∈ {1, ξ}N} ⊂ G∧ and since the correspondence h 7→ χh is injective, we get that Card G∧ ≥ c. The following statement shows that Proposition 7.2(b) is the best possible in the class of locally compact groups. Proposition 7.3. For an infinite locally compact Hausdorff topological abelian group X TFAE: (i) X is compact connected and metrizable. (ii) Card (c0 (X), u0 )∧ = ℵ0 . Proof. (i) =⇒ (ii) by Proposition 7.2(b). (ii) =⇒ (i). Let us see first that X is compact and metrizable. Write: G := (c0 (X), u0 ). Let ϕ : G → X be the first projection, i.e., the mapping which sends (xn )n∈N to x1 . Then ϕ∧ : X ∧ → G∧ is injective. So, Card G∧ = ℵ0 implies that Card X ∧ ≤ ℵ0 . From this by Proposition 2.2 we get that X is compact metrizable. Then by Proposition 7.2(c) the equality Card (c0 (X), u0 )∧ = ℵ0 implies that X is connected. The connected groups in the class of all compact metrizable abelian groups can be characterized also as follows: Proposition 7.4. For an infinite compact metrizable abelian group X TFAE: (i) X is connected.


18

(ii) (c0 (X), u0 )∧ = (X ∧ )(N) ; (iii) Card (c0 (X), u0 )∧ = ℵ0 . (iv) Card (c0 (X), u0 )∧ < c. Proof. (i) =⇒ (ii) =⇒ (iii) by Proposition 7.2(a). (iii) =⇒ (iv) is clear. (iv) =⇒ (i) by Proposition 7.2(c).

The following theorem provides a wide class of precompact metrizable groups which are not Mackey in LQC, extending thus the result of Proposition 5.3. Theorem 7.5. Let X be an infinite compact connected metrizable group. We have: ∧ ∧ (a) (c0 (X), p0 ) = (c0 (X), u0 ) . (b) u0 is a locally quasi-convex Polish group topology compatible for (c0 (X), p0 ). (c) (c0 (X), p0 ) is a precompact metrizable group which is not a Mackey group in LQC. Further, it is connected. ∧

∧

Proof. (a) Since p0 ≤ u0 , we have: (c0 (X), p0 ) ⊂ (c0 (X), u0 ) . To prove the converse inclusion, fix an ∧ arbitrary κ ∈ (c0 (X), u0 ) . By Proposition 7.4 there exists h = (ξn )n∈N ∈ (X ∧ )(N) such that κ = χh . Consequently, κ ∈ (c0 (X), p0 )∧ and (a) is proved. (b) u0 is a locally quasi-convex Polish group topology by Proposition 3.6. By (a) it is compatible for (c0 (X), p0 ). (c) (c0 (X), p0 ) is a precompact metrizable group because it is a topological subgroup of the compact metrizable group (X N , p). It is not a Mackey group in LQC because by (b) u0 is a locally quasi-convex group topology, compatible for (c0 (X), p0 ) and strictly finer than p0 (Proposition 3.2 (a1 ) ). Connectedness follows from Proposition 3.6 (c). 7.6. Groups with uncountable dual. The following statement shows that the topological group (SN , u) has a dual much ”bigger” than (c0 (S), u0 ). Proposition 7.7. Let X 6= {0} be a compact group, G = (X N , u). Then Card CHom(G, X) ≥ 2c . In particular, if G = S, then Card G∧ = 2c . Proof. Denote by F the set of all ultrafilters on N. It is known that Card F = 2c .

(7.3)

For a filter F on N, (xn )n∈N ∈ X N and x ∈ X we write: lim xn = x n,F

if for every W ∈ N (X) one has {n ∈ N : xn − x ∈ W } ∈ F . Since X is compact Hausdorff, it follows that for every F ∈ F and (xn )n∈N ∈ X N there exists a unique x ∈ X such that limn,F xn = x. For a filter F ∈ F define the mapping χF : X N → X by the equality: χF (x) = lim xn , n,F

∀x = (xn )n∈N ∈ X N .

Then χF ∈ CHom(X N , X) ∀F ∈ F .

(7.4) To verify (7.4), fix F ∈ F . As

χF (x + y) = lim(xn + yn ) = lim xn + lim yn = χF (x) + χF (y), n,F

n,F

n,F

∀x, y ∈ X N ,

we conclude that χF ∈ Hom(X N , X). To see that χ is continuous on (X N , u), fix a closed W ∈ N (X). Since W is closed, for x = (xn )n∈N ∈ W N we shall have χF (x) = lim xn ∈ W . n,F

N

Consequently, χF (W ) ⊂ W . From this relation, as W N ∈ N (X N , u), we get that χF is continuous on (X N , u) and (7.4) is proved.


19

We have also: (7.5)

F1 ∈ F, F2 ∈ F, F1 6= F2 =⇒ χF1 6= χF2

In fact, as F1 and F2 are distinct ultrafilters, there is F ∈ F1 such that F 6∈ F2 . Let x = (xn )n∈N ∈ X N be defined by conditions: xn = 0 if n ∈ F and xn = a 6= 0 if n ∈ N \ F . Then χF1 (x) = 0 and χF2 (x) = a. Therefore, χF1 6= χF2 and (7.5) is proved. Clearly (7.3),(7.4) and (7.5) imply that Card CHom(X N , X) ≥ 2c . 8. Open questions It follows from [9, Proposition 5.4] that every non-meager (G, µ) ∈ LCS is a Mackey group in LQC. Conjecture 8.1. Every metrizable (G, µ) ∈ LCS is a Mackey group in LQC. We do not even know if R(N) with the topology induced from the product space RN is a Mackey group in LQC. Remark 8.2. We conjecture that Proposition 5.2(c) remains true for a (not necessarily complete metrizable) topological abelian group. It is clear that if G is a discrete group, then G is a Mackey group in MAP. Conjecture 8.3. If G ∈ MAP is a Mackey group in MAP, then G is a discrete group. The conjecture 8.3 in terms of the Mackey topology can be reformulated as follows. Conjecture 8.4. If for a precompact topological group (G, ν) there exists the MAP-Mackey topology in G associated with ν, then ν is the finest precompact group topology in G. Remark 8.5. In [6] it is shown that a non-complete precompact metrizable (G, µ) can be a Mackey group in LQC. It is also known that every locally pseudocompact (G, µ) is a Mackey group in LQC (cf. [12]). An internal description of groups (G, τ ) ∈LQC which are Mackey in LQC is unknown. References [1] H.Anzai, and S. Kakutani, Bohr compactifications of a locally compact Abelian group. II. Proc. Imp. Acad. Tokyo 19, (1943), pp. 533–539. [2] Aussenhofer, L. Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, Diss. Math. CCCLXXXIV. Warsaw, 1999. [3] W.Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Mathematics, 1466. Springer-Verlag, Berlin, 1991. [4] M. Barr and H. Kleisli,, On Mackey topologies in topological abelian groups. Theory Appl. Categ. 8 (2001), pp. 54–62 [5] G. Bennett and N. J. Kalton, Inclusion theorems for K-spaces. Canad. J. Math. 25 (1973), pp. 511–524. 46A45 [6] F. G. Bonales, F. J. Trigos-Arrieta and R. Vera Mendoza, A Mackey-Arens theorem for topological abelian groups, Bol. Soc. Mat. Mexicana (3) 9 (2003), no. 1, pp. 79–88. [7] N. Bourbaki, General topology, I, Hermann, Paris VI, 1966. [8] N. Bourbaki, General topology, II, Hermann, Paris VI, 1966. [9] M. J. Chasco, E. Mart´ın-Peinador and V. Tarieladze, On Mackey Topology for groups, Stud. Math. 132, No.3, pp. 257-284 (1999). [10] W. W. Comfort and K. A. Ross, Topologies induced by groups of characters. Fund. Math. 55 (1964), pp. 283–291. [11] W. W. Comfort and K. A. Ross, Pseudocompactness and uniform continuity in topological groups. Pacific J. Math. 16 (1966) pp. 483–496. [12] L. De Leo, D. Dikranjan, E. Mart´ın-Peinador and V. Tarieladze, Duality theory for groups revisited: g-barrelled, Mackey and Arens groups. Preprint, 2009. [13] D. Dikranjan, Iv. Prodanov and L. Stoyanov, Topological Groups: Characters, Dualities and Minimal Group Topologies, Pure and Applied Mathematics, vol. 130, Marcel Dekker Inc., New York-Basel, 1989. [14] D. Dikranjan and D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on abelian groups. Topology Appl. 151 (2005), no. 1-3, pp. 2–54. [15] P. Enflo, Uniform structures and square roots in topological groups. Israel J. Math. 8 (1970), pp. 230-252. [16] S. S. Gabriyelyan, Groups of quasi-invariance and the Pontryagin duality. Arxiv preprint arXiv:0812.1671v6[Math.GN] 1oct. 2009 [17] J. Galindo and S. Garcia-Ferreira, Compact groups containing dense pseudocompact subgroups without non-trivial convergent sequences. Topology Appl. 154 (2007), no. 2, 476–490.

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[18] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I. Die Gr¨ uundlehren der Mathematischen Wissenschaften 115. Springer-Verlag 1963. [19] J.L. Kelley and I. Namioka, Linear topological spaces, Springer-Verlag, 1963. [20] J.W. Nienhuys, A solenoidal and monothetic minimally almost periodic group. Fund. Math. 73 (1971/72), no. 2, pp. 167–169. [21] S. Rolewicz, Some remarks on monothetic groups, Colloq. Math. XIII, ( 1964), pp. 28–29. [22] H. H. Schaefer, Topological vector spaces. The Macmillan Co., New York; Collier-Macmillan Ltd., London 1966 [23] L. J. Sulley, On countable inductive limits of locally compact abelian groups. J. London Math. Soc. (2) 5 (1972), pp. 629–637. [24] M. Tkachenko, Self-duality in the class of precompact groups. Topology Appl. 156 (2009), no. 12, pp. 2158–2165. [25] N. Th. Varopoulos, Studies in harmonic analysis, Proc. Cambridge Philos. Soc. 60 1964 pp. 465–516. [26] A. Weil, L’int´ egration dans les groupes topologiques et ses applications. (French) Actual. Sci. Ind., no. 869. Hermann et Cie., Paris, 1940. 158 pp [27] N. Ya. Vilenkin, The theory of characters of topological Abelian groups with boundedness given, Izvestiya Akad. Nauk SSSR. Ser. Mat. 15, (1951), pp. 439–162. Department of Mathematics and Computer Science, University of Udine, Via delle Scienze, 208-Loc. Rizzi, 33100 Udine, Italy E-mail address: [email protected] Departamento de Geometr´ıa y Topolog´ıa, Universidad Complutense de Madrid, 28040 Madrid, Spain E-mail address: em [email protected] Niko Muskhelishvili Institute of Computational Mathematics, 0171 Tbilisi, Georgia E-mail address: [email protected]